There are a number of formalisms in the literature that aim at providing a rigorous way of specifying partial algebraic structure. Freyd's essentially algebraic theories [1]} were introduced informally, but were subsequently formalised and generalised in various ways [2]}, [3]}, [4]}.
A different, but equally expressive approach is via finite limit sketches [2]}. Nevertheless, none of these approaches can claim to have the foundational status of classical equational theories - e.g. they do not, per se, provide a canonical notion of syntax to replace classical terms, nor a calculus for (partial) equational reasoning about the categories of models they define. Tout court, none of them can claim to be equational. Interestingly, the semantic landscape (i.e. the corresponding notion of partial variety) is better understood than the syntax. The class of models of essentially algebraic theories and finite limit sketches are closely related to Gabriel-Ulmer duality [6]}, which asserts a contravariant (bi)equivalence between the category of categories with finite limits and the category of locally finitely presentable (LFP) categories.
